Schauder Fixed-Point Theorem — 95 years later.

In 1930, J. Schauder claimed the fixed-point property of convex compacta in metric linear spaces. Surprisingly, in 1935, he posted a question of whether that claim held. Contemporary, Schauder’s claim is referred to as Schauder Conjecture. I will provide a list of false calls on settling the conjecture that I am aware of. Fixed-point vs absolute retract property of convex compacta will be emphasized.

I will concentrate on numerous recent attempts to prove the conjecture via -convexity, . Also, I will sketch major steps in Cauty’s approach published in Serdica Math. J. that are relevant to the case of convex compacta.

Regretfully, I can’t say that the conjecture has been settled.